Decoherence limitsquantum computationOr does it?

Robert Raussendorf,

University of British Columbia

Seven Pines,May 7, 2010

Outline

Fighting decoherence:

• Magic state distillation

Befriending decoherence:

• Measurement-based quantum computation

• Quantum computation by dissipation

Part I:

Quantum computation with magic states

S. Bravyi and A. Kitaev, Phys. Rev. A, 2005

Undoing decoherence

Capstone result:

Threshold theorem. Given [fill in a suitable error model], if the

error per elementary gate in a quantum computer is below a crit-

ical threshold, arbitrarily long and arbitrarily accurate quantum

computation is possible.

Q: What is the noise threshold?

Noise threshold for fault-tolerance

... exact value may be hard to calculate. Instead derive

• Upper bound: For a given set of computational primitives,if the noise level exceeds the upper bound, then no method,however clever, can achieve fault-tolerance.

• Lower bound: For a given set of computational primitives,if the noise level is less than the lower bound, then at leastone method makes the computation fault-tolerant.

noise level0 high

lower bound upper bound

QEC works for sure QEC fails for sure???

Quantum computation using magic states

We consider the computational primitives

{CNOT-gate,Hadamard-gate, |T 〉}

Therein,

|T 〉 =|0〉+ eiπ/4|1〉√

2. (1)

is the “magic” state.

Computational primitives - CNOT

• The CNOT gate is a two-qubit gate. It acts as

CNOTc,t = |0〉c〈0| ⊗ I(t) + |1〉c〈1| ⊗ σ(t)x . (2)

• The CNOT is the only computational primitive in the set

which has the power to entangle.

Computational primitives - Hadamard

• The Hadamard gate H acts as

H = |+〉〈0|+ |−〉〈1|, (3)

where |±〉 := 1/√

2(|0〉 ± |1〉).

• The Hadamard gate rotates the Bloch sphere of a qubit byan angle of π about the axis in the middle between x and z.

z

x y

H

Computational primitives - magic state |T 〉

• State |T 〉 = |0〉+eiπ/4|1〉√2

implements gate T = exp(iπ/8Z).

Z

T-1TXT

outcome +1outcome -1

good

random walk on a circle

• Gate construction is probabilistic. Repeat until success.

• The used primitives are universal for quantum computation!Any unitary in U ∈ SU(2n), for any n, can be built as a sequence of the

above gates.

Decoherence model

Computational primitive Quality

CNOT-gate perfect

Hadamard-gate perfect

magic state |T 〉 noisy

• Instead of pure states |T 〉 have states ρT ≈ |T 〉〈T |. Use

fidelity F (ρT ) =√〈T |ρT |T 〉 as measure for quality.

• Motivation for this noise model: certain versions of topo-

logical quantum computation which are non-universal.

Bravyi & Kitaev’s results

Result 1. If the noisy magic states ρT are in-

side the octahedron P8 inscribed in the Bloch

sphere, then quantum computation using the

primitives {CNOT, H, ρT} can be efficiently

classically simulated.

z

x y

P8

Result 2 [Magic state distillation]. If the noisy magic states ρTare such that F (ρT ) ≥ 0.927, then arbitrarily long and accurate

universal quantum computation is possible with the primitives

{CNOT, H, ρT}.

Derivation of Result 1

• How powerful is the gate set {CNOT, H, S = exp(2×iπ/8Z)}?

• Not powerful at all. It is efficiently classically simulatable.

First, consider the one-qubit gates H, S = exp(iπ/4Z):

z

x y

HS180 degs

90 degs

• H, S leave the octahedron invariant.

• H, S generate the octahedral group.

Not dense in SU(2), hence no 1-

qubit universality.

Heisenberg picture

Consider the action of H, S = exp(iπ/4Z), CNOT on Pauli

operators:

z

x y

HS180 degs

90 degs

• HXH† = Z, HZH† = X.

• SXS† = Y , SZS† = Z.

• CNOTX(c)CNOT † = X(c) ⊗X(t) ...

• Above gates map Pauli operators onto Pauli operators.

⇒ Evolution easily trackable in Heisenberg picture.

Gottesman-Knill theorem

⇒ Evolution easily trackable in Heisenberg picture. Leads to

Gottesman-Knill Theorem:

Theorem 1. Quantum computation with {CNOT, H, S}, on ini-

tial qubit states |0/1〉, |±〉, |±y〉, and with readout measurements

in the X, Y or Z-basis can be efficiently classically simulated.

xy

z

Result 1 of Bravyi & Kitaev

xy

z

• The only computational primitive that evades the

Gottesman-Knill theorem is the magic state |T 〉.

• Can the noisy state ρT be described as a probabilistic mix-

ture of {|0,1〉, |±〉, |±y〉}?

• If yes, then the computation can be efficiently simulated us-

ing the Gottesman-Knill theorem + Monte Carlo sampling.

Result 1. If the noisy magic states ρT are inside the octahedron P8 in-scribed in the Bloch sphere, then quantum computation using the primi-tives {CNOT, H, ρT} can be efficiently classically simulated.

Result 2: Magic state distillation

ρT

ρT

ρT

ρT

ρT

ρTm

agic

sta

te d

isti

llati

on

In: 15 copies,low-F

Out: 1 copy,high-F

0.50.5

1

1

distillationthreshold

FIn

OutF

Result 2 [Magic state distillation]. If the noisy magic states ρT are suchthat F (ρT) ≥ 0.927, then arbitrarily long and accurate universal quantumcomputation is possible using the primitives {CNOT, H, ρT}.

Results 1 & 2

F ( )ρT < 0.924:efficient classical simulation

F ( )ρT < 0.927:Universal QC

X/Y - equator of the Bloch sphere

But where’s the magic?

X

Y

Part II:

Measurement-based quantum computation

R. Raussendorf and H.J. Briegel, PRL 86, 5188 (2001).

Measurement-based Quantum Computation

Z

X

Y

Z

X

Y

Unitary transformation Projective measurement

deterministic,reversible

probabilistic,irreversible

p

1-p

The one-way quantum computer

measurement of Z (�), X (↑), cosαX + sinαY (↗)

• Universal computational resource: cluster state.

• Information written onto the cluster, processed and

read out by one-qubit measurements only.

Trading entanglement for output

measurement

yet unmeasuredpart of cluster

enta

ng

lem

ent

time

• Intuition: Entanglement = Resource

Part III:

Quantum computation by dissipation

F. Vertraete, M. Wolf and J.I. Cirac, Nature Phys. 5 (2009).

Computing fridges

• Cooling into the ground state of a simple (3-body, say)

Hamiltonian were an incredibly powerful computational

tool ...

If one could avoid local minima.

V

• Bold task: Solve NP-complete problems by cooling.

• Task: Universal quantum computation by cooling.

Computing fridges

Result 3. Consider a quantum circuit of n qubits and T gates,|Ψout〉 = UT UT−1 ..U2U1|0〉. The output of this quantum com-putation can be efficiently simulated by local dissipative evolutionon n+ T qubits.

*

Why n+ T qubits?

Total Hilbert space H = HQ-register︸ ︷︷ ︸n qubits

⊗ Hclock︸ ︷︷ ︸T qubits

.

*: Image adapted from Nature Physics.

Computing fridges

Consider dissipative evolution described by Lindblad equation

with a Liouville operator

L(ρ) =∑k

LkρL†k −

1

2

{L†kLk, ρ

}+, (4)

where

Li = |0〉i〈1| ⊗ |0〉〈0|, ∀i = 1..n, (initialize QR)

Lt = Ut ⊗ |t+ 1〉〈t|+ h.c., ∀t = 1..T, (advance clock)(5)

Computing fridges

The above dissipative evolution has the following properties

1. Unique fixpoint is a history state

ρ0 =1

T + 1

∑t

|ψt〉〈ψt| ⊗ |t〉〈t|, (6)

where |ψ〉t = state of QR at time t.

2. Liouville operator has a spectral gap ∆ ∼ 1/T2.

Good approximation to ρ0 is reached in poly time τ ∼ T2.

Final step of the computation: After evolution for time τ , mea-sure the clock register. If obtain t = T then read out |Ψout〉.Otherwise start over.

Why decoherence did not hurt in Ex. II, III?

•

Closing remark on entanglement

Why did decoherence not hurt in Ex. II, III ?

Because we only depleted

coherences

that we didn’t care about.

Example III - Universal AQC vs. DQC

Adiabatic QC (unitary) Dissipative QC

H(t)=H (1-t) + H tI F

initial Hamiltonian:GS easy to prepare

final Hamiltonian:encodes comp. resultin its ground state

HI, HF exist such that1

1. GS is history state:

|Ψ〉 = 1√T+1

∑t |ψt〉|t〉

2. Min gap ∼ 1/T2.

L exists such that

1. FP is history state:

ρ0 = 1T+1

∑t |ψt〉〈ψt| ⊗ |t〉〈t|

2. Gap ∼ 1/T2.

1: D. Aharonov et al., arXiv:quat-ph/0405098 (2004).

Example III - Universal AQC vs. DQC

Adiabatic QC:

HI, HF exist such that

1. GS is history state:

|Ψ〉 = 1√T+1

∑t |ψt〉|t〉

2. Min gap ∼ 1/T2.

Dissipative QC:

L exists such that

1. FP is history state:

ρ0 = 1T+1

∑t |ψt〉〈ψt| ⊗ |t〉〈t|

2. Gap ∼ 1/T2.

• Adiabatic QC: history state is a coherent superposition,

Dissipative QC: history state is a mixture.

• Recall: Measure clock at end of computation.

⇒ Coherence between clock states is not important.

Example II - Measurement-based QC

Instead of the one-way QC, look at simpler example:

Z

T TX S

ρ

Recall: S = exp(iπ/4Z), T = exp(iπ/8Z)

• Decohered is only the post-measurement state or the lower

qubit, which we discard.

• Again, decoherence does not affect the computational de-

grees of freedom.

Remark on entanglement

Do the dissipative evolutions discussed in Examples II, III drive

the respective system to a “classical” state?

• One-way QC: Yes. Final state is a product state.

• Dissipative QC: No. Final state is highly entangled∗.

*: The entanglement of the history state ρ0 equals the time-averaged entan-

glement of the circuit quantum register.